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**Complexity of inheritance of F-convexity for restricted**

**games induced by minimum partitions**

### Alexandre Skoda

**To cite this version:**

Alexandre Skoda. Complexity of inheritance of F-convexity for restricted games induced by minimum partitions. 2016. �halshs-01382502�

**Documents de Travail du **

**Centre d’Economie de la Sorbonne **

**Complexity of inheritance of F-convexity for **

**Complexity of inheritance of F-convexity for**

**restricted games induced by minimum partitions **

### Alexandre S

KODA### Complexity of inheritance of

### F-convexity for

### restricted games induced by minimum partitions.

A. Skoda∗ July 18, 2016

Abstract

Let G = (N, E, w) be a weighted communication graph (with weight function w on E). For every subset A ⊆ N , we delete in the sub-set E(A) of edges with ends in A, all edges of minimum weight in E(A). Then the connected components of the corresponding induced subgraph constitute a partition of A that we call Pmin(A). For

ev-ery game (N, v), we define the Pmin-restricted game (N, v) by v(A) =

P

F ∈Pmin(A)v(F ) for all A ⊆ N . We prove that we can decide in

poly-nomial time if there is inheritance of F-convexity from (N, v) to the Pmin-restricted game (N, v) where F-convexity is obtained by

restrict-ing convexity to connected subsets.

Keywords: communication networks, cooperative game, convex game, restricted game, partitions, supermodularity, graph, complexity.

AMS Classification: 91A12, 91A43, 90C27, 05C75, 68Q25.

### 1

### Introduction

We consider, on a given finite set N with |N | = n, a weighted communication
*structure, i.e., a weighted gragh G = (N, E, w) where w is a weight function*
defined on the set E of edges of G. For a given subset A of N , we denote
by E(A) the set of edges in E with both endvertices in A, and by Σ(A) the
subset of edges of minimum weight in E(A). Let GA be the graph induced

*by A, i.e., G*A = (A, E(A)). Then ˜GA = (A, E(A) \ Σ(A)) is the graph

obtained by deleting the minimum weight edges in GA. In (Skoda, 2016) we

introduced the correspondence Pmin on N which associates to every subset

A ⊆ N , the partition Pmin(A) of A into the connected components of ˜GA.

Then for every game (N, v) we defined the restricted game (N, v) associated
∗_{Corresponding author. Universit´e de Paris I, Centre d’Economie de la Sorbonne,}
106-112 Bd de l’Hˆopital, 75013 Paris, France. E-mail: alexandre.skoda@univ-paris1.fr

with Pmin by:

(1) v(A) = X

F∈Pmin(A)

v(F ), for all A ⊆ N.

We more simply refer to this game as the Pmin-restricted game. v is the

characteristic function of the game, v : 2N _{→ IR, A 7→ v(A) and satisfies}

v(∅) = 0. Compared to the initial game (N, v), the restricted game (N, v) takes into account the combinatorial structure of the graph and the ability of players to cooperate in a given coalition and therefore differents aspects of cooperation restrictions. In particular, assuming that the edge weights re-flect the degrees of relationships between players, Pmin(A) gives a partition

of a coalition A into subgroups where players are in privileged relationships (with respect to the minimum relationship degree in A). Many other corre-spondences have been considered to define restricted games (see, e.g., Myer-son (1977); Algaba et al. (2001); Bilbao (2000, 2003); Faigle (1989); Grabisch and Skoda (2012); Grabisch (2013)). For a given correspondence a classical problem is to study the inheritance of basic properties as superadditivity and convexity from the underlying game to the restricted game. Inheritance of convexity is of particular interest as it implies that good properties are inherited, for instance the non-emptyness of the core, and that the Shapley value is in the core. For the Pmincorrespondence we proved in (Grabisch and

Skoda, 2012) that we always have inheritance of superadditivity from (N, v)
to (N, v). Let us observe that inheritance of convexity is a strong
prop-erty. Hence it would be useful to consider weaker properties than convexity.
Following alternative definitions of convexity in combinatorial optimization
and game theory when restricted families of subsets are considered (not
*necessarily closed under union and intersection), see, e.g., (Edmonds and*
Giles, 1977; Faigle, 1989; Fujishige, 2005) we introduced in (Grabisch and
Skoda, 2012) the F-convexity by restricting convexity to the family F of
connected subsets of G. In (Skoda, 2016), we have characterized inheritance
of F-convexity for Pmin by five necessary and sufficient conditions on the

edge-weights. Of course the study of inheritance of F-convexity is also a first key step to characterize inheritance of convexity in the general case. We have also highlighted in (Skoda, 2016) a relation between Myerson’s re-stricted game introduced in (Myerson, 1977) and the Pmin-restricted game.

Myerson’s restricted game (N, vM) is defined by vM(A) =P

F∈PM(A)v(F ) for all A ⊆ N , where PM(A) is the set of connected components of GA.

We proved that inheritance of convexity for the Myerson’s game is equiv-alent to inheritance of F-convexity for the Pmin-restricted game associated

with a weighted graph with only two different weights. In the present pa-per we prove that inheritance of F-convexity can be checked in polynomial time for Pmin. Of course, directly testing inheritance of convexity (resp.

F-convexity) for the Pmin-correspondence is highly non polynomial. We

to compute the Pmin-restricted game (N, v) and to verify the convexity

*(resp. F-convexity) of (N, v), i.e., for all subsets A, B ∈ 2*N (resp. for
all subsets A, B such that A, B, and A ∩ B are connected), to verify that
v(A ∪ B) + v(A ∩ B) ≥ v(A) + v(B). The characterization given in (Skoda,
2016) implies that we can verify inheritance of F-convexity by checking that
some specific distributions of edge-weights are satisfied on all stars, paths,
cycles, pans1_{, and adjacent cycles of the graph G. Of course directly }

check-ing all these conditions would also lead to a non-polynomial algorithm as the number of paths and cycles can be exponential. We prove that we only have to take into account a polynomial number of specific paths and cycles and therefore we can decide in O(n6) time whether there is inheritance of F-convexity.

Moreover, to establish this result we have to more deeply analyze the relations between the star, path, cycle, pan and adjacent cycles conditions characterizing inheritance of the F-convexity. It is of independent interest as we prove that some of them are not completely independent of the others and we highlight their relations which for the sake of simplicity were only implicitely contained in (Skoda, 2016). In particular we observe that the cycle condition in (Skoda, 2016) splits into an intermediary cycle condition which is a consequence of the star and path conditions and a condition on chords of the cycles. Also a part of the pan condition is a consequence of star and path conditions. We prove that the cycle condition is a consequence of the star, path and adjacent cycles conditions. Hence inheritance of F-convexity can be characterized by only four of the previous conditions: the star, path, pan, and adjacent cycles conditions.

Star condition can easily be checked in polynomial time. Checking the
other conditions in polynomial time requires a deeper and careful analysis.
In a first fundamental step, we prove that to verify path condition in
poly-nomial time we only have to consider one minimum weight spanning tree
T in G and to verify path condition for the paths joining the end vertices
of T and to test intermediary cycle condition for the fundamental cycles of
T . Then the main step, which is also the most difficult one, deals with the
*adjacent cycles condition which a priori involves any pair of adjacent cycles*
(C, C′). A first contradiction to the adjacent cycle condition is the existence
of two cycles C anc C′ with two common adjacent edges e1 and e2 of non

maximum weight. We prove that if there exist two such cycles then there also exist two specific cycles ˜C and ˜C′ with two common edges of non max-imum weight which are associated with shortest paths in some particular subgraphs depending only on the pair (e1, e2). As a result of their features

such cycles ˜C and ˜C′ can be detected in polynomial time. A similar result holds for the second possible contradiction to the adjacent cycle condition which corresponds to the existence of two adjacent cycles C and C′ with

only one common non maximum weight edge e1. Hence, looking for ˜C and

˜

C′ for any pair of adjacent edges (resp. for any edge) in G, we can detect any contradiction to the adjacent cycles condition. Therefore the adjacent cycles condition can be checked in polynomial time. Assuming that the star, path and adjacent cycles condition are satisfied, the cycle condition is necessar-ily satisfied. Finally using the cycle condition we establish a new condition equivalent to the pan condition that we can easily verify in polynomial time using an appropriate shortest path.

The article is organized as follows. In Section 2, we give preliminary defi-nitions and results. In particular, we recall the defidefi-nitions of convexity and F-convexity. In section 3, we recall necessary and sufficient conditions on graph edge-weights to have inheritance of F-convexity for the correspon-dence Pmin established in (Skoda, 2016). Then section 4 is the main part

of this article where we establish that inheritance of F-convexity for the Pmin-correspondence can be checked in polynomial time.

### 2

### Preliminary definitions and results

Let N be a given set. We denote by 2N the set of all subsets of N . A
*game (N, v) is zero-normalized if v(i) = 0 for all i ∈ N . A game (N, v)*
*is superadditive if, for all A, B ∈ 2*N _{such that A ∩ B = ∅, v(A ∪ B) ≥}

v(A) + v(B). For any given subset ∅ 6= S ⊆ N , the unanimity game (N, uS)

is defined by:

(2) uS(A) =

1 if A ⊇ S, 0 otherwise. We note that uS is superadditive for all S 6= ∅.

Let us consider a game (N, v). For arbitrary subsets A and B of N , we define the value:

∆v(A, B) := v(A ∪ B) + v(A ∩ B) − v(A) − v(B).

*A game (N, v) is convex if its characteristic function v is supermodular,*

*i.e., ∆v(A, B) ≥ 0 for all A, B ∈ 2*N_{. We note that u}

S is supermodular for

*all S 6= ∅. Let F be a weakly union-closed family* 2 of subsets of N such
that ∅ /∈ F. A game v on 2N _{is said to be F-convex if ∆v(A, B) ≥ 0, for}

all A, B ∈ F such that A ∩ B ∈ F. If F = 2N _{\ {∅} then F-convexity}

corresponds to convexity.

For a given graph G = (N, E), we say that a subset A ⊆ N is connected if the induced graph GA= (A, E(A)) is connected. In this paper F will be

the family of connected subsets of N . 2

F is weakly union-closed if A ∪ B ∈ F for all A, B ∈ F such that A ∩ B 6= ∅ (Faigle et al., 2010). Weakly union-closed families were introduced and analysed in Algaba (1998); Algaba et al. (2000) and called union stable systems.

We recall the following theorems established in (Skoda, 2016).

Theorem 1. *Let G = (N, E, w) be an arbitrary weighted graph, and P an*
*arbitrary correspondence on N . The following claims are equivalent:*

*1) For all ∅ 6= S ⊆ N , u*S *is superadditive.*

*2) For all subsets A ⊆ B ⊆ N , P(A) is a refinement of the restriction of*

*P(B) to A.*

*3) For all superadditive game (N, v) the P-restricted game (N, v) is *
*super-additive.*

Theorem 2. *Let G = (N, E, w) be an arbitrary weighted graph, and P an*

*arbitrary correspondence on N . If for each non-empty subset S ⊆ N , u*S *is*

*superadditive, then the following claims are equivalent.*

*1) For all non-empty subset S ⊆ N , the game (N, u*S*) is F-convex.*

*2) For all i ∈ N and for all A, B ∈ F, A ⊆ B ⊆ N \ {i} such that A ∪ {i} ∈*

*F, we have for all A*′ *∈ P(A ∪ {i}), P(A)*_{|A}′ = P(B)_{|A}′*.*

Let G = (N, E, w) be a weighted graph. We set |N | = n and |E| = m. We assume that all weights are strictly positive and denote by wkor wij the

weight of an edge ek= {i, j} in E. The adjacency matrix A of G is an n × n

matrix defined by:

aij =

wij if {i, j} ∈ E,

0 otherwise.

In the next section we recall necessary and sufficient conditions on the weight vector w established in (Skoda, 2016) for the inheritance of F-convexity from the original communication game (N, v) to the Pmin-restricted

game (N, v).

### 3

### Necessary conditions on edge-weights

A star Sk corresponds to a tree with one internal vertex and k leaves. We

consider a star S3 with vertices 1, 2, 3, 4 and edges e1 = {1, 2}, e2 = {1, 3}

and e3 = {1, 4}.

Star Condition. *For every star of type S*3 *of G, the edge-weights*

w1, w2, w3 *satisfy, after renumbering the edges if necessary:*

w1 ≤ w2 = w3.

Proposition 3. *Let G = (N, E, w) be a weighted graph. If for every *

*unani-mity game (N, u*S*), the P*min*-restricted game (N, u*S*) is F-convex, then the*

Path Condition. *For every elementary path γ = (1, e*1, 2, e2, 3, . . . , m,

em*, m + 1) in G and for all i, j, k such that 1 ≤ i < j < k ≤ m, the*

*edge-weights satisfy:*

wj ≤ max(wi, wk).

Proposition 4. *Let G = (N, E, w) be a weighted graph. If for every *

*una-nimity game (N, u*S*), the P*min*- restricted game (N, u*S*) is F-convex, then*

*the Path Condition is satisfied.*

For a given cycle in G, C = {1, e1, 2, e2, . . . , m, em, 1} with m ≥ 3, we

denote by E(C) the set of edges {e1, e2, . . . , em} of C and by ˆE(C) the set

composed of E(C) and of the chords of C in G.

Weak Cycle Condition. *For every simple cycle of G, C =*

{1, e1, 2, e2, . . . , m, em*, 1} with m ≥ 3, the edge-weights satisfy, after*

*renumbering the edges if necessary:*

(3) w1 ≤ w2≤ w3 = · · · = wm = M

*where M = max*_{e∈E(C)}*w(e).*

Intermediary Cycle Condition.

*1) Weak Cycle condition.*

*2) Moreover w(e) ≤ w*2 *for all chord incident to 2, and w(e) ≤ ˆ*M =

max_{e∈ ˆ}_{E(C)} *for all chord non incident to 2. Moreover:*

*• If w*1 ≤ w2 < ˆ*M then w(e) = ˆM for all e ∈ ˆE(C) non-incident*

*to 2. If e is a chord incident to 2 then w*1≤ w2 = w(e) < ˆ*M or*

w(e) < w1 = w2< ˆ*M .*

*• If w*1 < w2= ˆ*M , then w(e) = ˆM for all e ∈ ˆ*E(C) \ {e1*}.*

Cycle Condition.

*1) Weak Cycle condition.*

*2) Moreover w(e) = w*2 *for all chord incident to 2, and w(e) = ˆM for*

*all e ∈ ˆE(C) non incident to 2.*

Proposition 5. *Let G = (N, E, w) be a weighted graph.*

*1. If G satisfies the Path condition then the Weak Cycle condition is*
*satisfied.*

*2. If G satisfies the Star and Path conditions then the Intermediary Cycle*
*condition is satisfied.*

*3. If for every unanimity game (N, u*S*), the P*min*- restricted game (N, u*S)

*is F-convex, then the Cycle Condition is satisfied.*

Weak Pan Condition. *For all connected subgraphs corresponding*

*to the union of a simple cycle C = {e*1, e2, . . . , em*} with m ≥ 3,*

*and an elementary path P such that there is an edge e in P with*

w(e) ≤ min1≤k≤mwk *and |V (C) ∩ V (P )| = 1, the edge-weights satisfy:*

*(a) either w*1 = w2 = w3= · · · = wm = ˆ*M ,*

*(b) or w*1= w2 < w3 = · · · = wm= ˆ*M ,*

*where ˆ*M = max_{e∈ ˆ}_{E(C)}*w(e). In this last case V (C) ∩ V (P ) = {2}.*

Pan Condition.

*1) Weak Pan condition.*

*2) If Claim (b) of the Weak Pan condition is satisfied and if moreover*

w(e) < w1 *then {1, 3} is a maximum weight chord of C.*

Proposition 6. *Let G = (N, E, w) be a weighted graph.*

*1) If G satisfies the Star, and Path conditions then the Weak Pan condition*
*is satisfied.*

*2) If for every unanimity game (N, u*S*), the P*min*-restricted game (N, u*S*) is*

*F-convex, then the Pan Condition is satisfied.*

We recall the proof of Proposition 6 as we will need it to prove a new formulation of the Pan condition in Section 4.

*Proof. 1) Let us consider C = {1, e*1, 2, e2, 3, . . . , m, em, 1}, and P = {j, em+1,

m + 1, em+2, m + 2, . . . , em+r, m + r} with j ∈ {1, . . . , m}, as represented in

Figure 1. We can assume w.l.o.g. that e = em+r (restricting P if necessary)

and that wm+j > wm+r = w(e) for all 1 ≤ j ≤ r − 1. Applying Claim 2

m j 1 2 e1 em e2 ej em+1 em+2 em+r

Figure 1: Pan formed by the union of C and P .

necessary:

(4) w1 ≤ w2≤ w3 = · · · = wm = ˆM .

Let us first assume 3 ≤ j ≤ m. Applying Proposition 4 to the path {2, e1, 1, em, m, . . . , j + 1, ej, j, em+1, m + 1, . . . , m + r − 1, em+r, m + r}, we

have wj ≤ max(w1, w(e)) = w1. Then (4) implies (a).

Let us now assume j ∈ {1, 2}. If r = 1 then wm+1 = w(e) ≤ w1.

Oth-erwise, applying Proposition 4 to the path {e1, em+1. . . , em+r}, we have

wm+1≤ max(w1, w(e)) = w1.

If j = 1, Proposition 3 applied to the star defined by {e1, em, em+1}, implies

wm+1≤ w1 = wm. Hence (4) still implies (a).

If j = 2, Proposition 3 applied to the star defined by {e1, e2, em+1}, implies

wm+1 ≤ w1 = w2. If w1 = w2 = ˆM then (a) is satisfied. Otherwise we

have w1 = w2 < ˆM and (b) is satisfied. Hence the Weak Pan condition is

satisfied.

2) Now if Claim (b) of the Weak Pan condition is satisfied, let us assume by contradiction that {1, 3} /∈ E(C). Claim 3 of Proposition 5 implies w(e) = ˆM (resp. w(e) = w2) for any chord e of C non incident (resp.

in-cident) to 2. Therefore we can assume w.l.o.g. that C has no maximum weight chord (otherwise we can replace C by a smaller cycle which still contains the vertices 1, 2, 3). Let us consider i ∈ V (C) \ {1, 2, 3}, A = V (C) \ {i} and B = A ∪ V (P ) as represented in Figure 2. Then Pmin(A) =

3
i 2
1
em
ei
e_{i−1}
e1
e2
e3
em+1 em+2 em+r
A
B
Figure 2: wm+r< w1= w2 < w3 = · · · = wm= M .

{{2}, {3, 4, . . . , i − 1}, {i + 1, . . . , m, 1}}, Pmin(A ∪ {i}) = {A ∪ {i} \ {2}, {2}},

and Pmin(B) = {B \ {m + r}, {m + r}} or Pmin(B) = {B} (this last case can

occur if there exists an edge e′ in GB with w(e

′

) < w(e) or with m + r as an end-vertex and w(e′) > w(e)). Therefore A′ := A∪{i}\{2} ∈ Pmin(A∪{i}),

but Pmin(B)_{|A}′ = {A \ {2}} 6= P_{min}(A)_{|A}′ and it contradicts Theorem 2.
Lemma 7. *Let G = (N, E, w) be a weighted graph satisfying the Star and*

*Path conditions. Then for all pairs (C, C*′*) of adjacent simple cycles in G,*

*we have:*

(5) M = maxˆ

e∈ ˆE(C)

w(e) = max

e∈E(C)w(e) = maxe∈E(C′_{)}w(e) = max

e∈ ˆE(C′)

*Proof. We first consider M = max*_{e∈E(C)}w(e) and M′ = max_{e∈E(C}′_{)}w(e).
Let us consider two adjacent cycles C and C′ with M < M′. There is at
least one edge e1 common to C and C′. Then we have w1 ≤ M < M′

and therefore e1 is a non-maximum weight edge in C

′

. By Claim 1 of
Proposition 5 the Weak Cycle condition is satisfied. It implies that there
are at most two non-maximum weight edges in C′. Therefore there exists
an edge e′_{2} in C′ adjacent to e1 with w

′

2 = M

′

. As M′ > M , e′_{2} is not an
edge of C. Let e2 be the edge of C adjacent to e1 and e

′

2. Then we have

w2 ≤ M < M′ but it contradicts the Star condition applied to {e1, e2, e′2}

(two edge-weights are strictly smaller than w′_{2}). Therefore M = M′. Finally
by Claim 2 of Proposition 5 the Intermediary Cycle condition is satisfied and
we have ˆM = M = M′ = ˆM′.

Adjacent Cycles Condition. *For all pairs (C, C*′*) of adjacent simple*

*cycles in G such that:*

*(a) V (C) \ V (C*′*) 6= ∅ and V (C*′*) \ V (C) 6= ∅,*

*(b) C has at most one non-maximum weight chord,*

*(c) C and C*′ *have no maximum weight chord,*

*(d) C and C*′ *have no common chord,*

*then C and C*′ *cannot have two common non-maximum weight edges.*

*Moreover C and C*′ *have a unique common non-maximum weight edge*

e1 *if and only if there are non-maximum weight edges e*2∈ E(C) \ E(C′)

*and e*′_{2} ∈ E(C′*) \ E(C) such that e*1, e2, e

′

2 *are adjacent and:*

• w1= w2 = w′2 *if |E(C)| ≥ 4 and |E(C*

′
*)| ≥ 4.*

• w1= w2 ≥ w′2 *or w*1= w2′ ≥ w2 *if |E(C)| = 3 or |E(C*′*)| = 3.*

Proposition 8. *Let G = (N, E, w) be a weighted graph. If for every *

*una-nimity game (N, u*S*) the P*min*-restricted game (N, u*S*) is F-convex, then the*

*Adjacent Cycles Condition is satisfied.*

Finally the following characterization of inheritance of F-convexity was established in (Skoda, 2016).

Theorem 9. *Let G = (N, E, w) be a weighted graph. For every *

*superaddi-tive and F-convex game (N, v), the P*min*-restricted game (N, v) is F-convex*

*if and only if the Path, Star, Cycle, Pan, and Adjacent cycles conditions are*
*satisfied.*

### 4

### Complexity of inheritance of

### F-convexity with

### P

minWe now prove that to verify Star, Path, Cycle, Pan, and Adjacent Cycles conditions we only have to consider a polynomial number of paths, stars, pans and cycles. Therefore we can build a polynomial algorithm to decide for a given weighted graph if there is inheritance of F-convexity from underlying games to Pmin-restricted games. We assume that graphs are represented by

their adjacency matrices.

Proposition 10. *Let G = (N, E, w) be a weighted graph. Star condition*

*can be verified in O(n*2_{) time.}

*Proof. For a given node i in N we denote by E*i the set of edges incident

to i. If |Ei| ≥ 3 then we select two edges e1, e2 in Ei and replace Ei by

Ei\ {e1, e2}. If w1 < w2 we set min = w1, max = w2, otherwise min = w2,

max = w1. Then we apply the following procedure. While Ei 6= ∅ we select

an edge e ∈ Ei. If min = max then if w(e) ≤ max we set min = w(e),

otherwise we stop. Otherwise we have min < max and if w(e) 6= max we stop. Otherwise we replace Ei by Ei \ {e}. If the procedure stops with

Ei 6= ∅, then there is a contradiction to the Star condition. Otherwise the

Star condition is satisfied for the star centered in i. Hence Star condition can
be verified for the star centered in i in O(n) time. As we have to repeat this
procedure for all nodes in N we can verify Star condition in O(n2_{) time.}

We now assume that the Star condition is satisfied by G.

*Adding an edge to a spanning tree creates a unique cycle, called a *

*fun-damental cycle. Let T be a minimum weight spanning tree of G and γ =*

{1, e1, 2, e2, . . . , m, em, m+1} be an elementary path of G. Let ej1, ej2, . . . , ejk, with 1 ≤ j1 < j2 < . . . < jk ≤ m, be the edges in E(γ) \ E(T ). For every

ejl ∈ E(γ) \ E(T ), we denote by Cl the fundamental cycle in G associated with T and ejl.

Remark 1. Of course two fundamental cycles associated with a spanning tree can be adjacent.

We suppose that G satisfies the Star condition and that every funda-mental cycle associated with T satisfies the Weak Cycle condition. (Note that we only have to check for each fundamental cycle that there are at most two edges of non-maximum weight and that these edges are adjacent. And there are exactly m − n + 1 fundamental cycles associated with T . ) For each fundamental cycle Cl, we denote by Ml its maximum edge weight.

As T is a minimum weight spanning tree we have wjl = Ml ≥ w(e) for all e ∈ E(Cl). Let γl be the restriction of γ to Cl and vl, v

′

l its end-vertices.

Let γ_{l}′ := Cl\ γl be the other path in Cl linking vl and v

′

l as represented in

vl _{e} v′_{l}
jl

Cl

γl

γ_{l}′

Figure 3: Paths γl and γ

′

l in Cl.

Note that γ′_{l} contains at least one edge otherwise vl = v

′

l and γ is not

an elementary path. We denote by γl,l+1 the part of γ linking v

′

l to vl+1.

γl,l+1 can be reduced to one vertex (if v

′

l = vl+1). We denote by γ0,1 (resp.

γk,m+1) the part of γ linking 1 to v1 (resp. v

′

k to m + 1). γ0,1 (resp. γk,m+1)

can also be reduced to one vertex. Then ˜γ = γ0,1 ∪ γ

′ 1∪ γ1,2 ∪ γ ′ 2 ∪ · · · ∪ γk−1,k ∪ γ ′

k ∪ γk,m+1 corresponds to a path in T linking 1 to m + 1. Let

us observe that if two successive fundamental cycles Cl, Cl+1 are adjacent

then γl′∪ γl,l+1∪ γ

′

l+1 is not a simple path (γl,l+1 is reduced to one vertex

but γ_{l}′ and γ_{l+1}′ have common edges). In this case we delete the common
edges to get a simple path (in fact we get a unique elementary path in T ) as
represented in Figures 4 and 5. Keeping the same notations for simplicity,
we now consider that ˜γ is an elementary path in T linking 1 to m + 1.

v′_{l}
vl
v_{l+1}′
e_{jl} =vl+1 e_{jl+1}
γ′_{l} γ_{l+1}′
Cl+1
Cl
Figure 4: We replace γl′∪ γl,l+1∪ γ
′
l+1 by (γl′∪ γ
′
l+1) \ (γ
′
l ∩ γ
′
l+1).
v′_{l}
vl
v_{l+1}′
e_{jl} =vl+1 e_{jl+1}
γ_{l}′
γ_{l+1}′
Cl+1
Cl

Figure 5: We can have γ_{l}′ ⊂ γ_{l+1}′ .

Lemma 11. *If C*l *and C*l+1 *are adjacent then M*l= Ml+1*.*

*Proof. As the Star condition is satisfied and as the fundamental cycles *

asso-ciated with T satisfy the Weak Cycle condition, the result is an immediate consequence of Lemma 7 page 8.

Lemma 12. *Let G = (N, E, w) be a weighted graph satisfying the Star*
*condition. Let us consider a minimum weight spanning tree T and an *

*ele-mentary path γ = {1, e*1, 2, e2, . . . , m, em*, m + 1} in G. Let us assume that*

*every fundamental cycle associated with T satisfies the Weak Cycle *

*condi-tion. Then for every fundamental cycle C*l *associated with T and e*jl *in*

*E(γ) \ E(T ), with 1 ≤ l ≤ k, either M*l= w1 *or w*m *or there exists an edge*

e ∈ E(˜*γ) such that M*l*= w(e).*

*Proof. Let us recall that w*jl = Ml.

Case 1 Let us consider Ck and let us assume v

′

k*= m + 1, i.e., C*k incident

to m + 1. Then em ∈ E(γk). If ejk = em or if wm = Mk the result is satisfied. Hence we can assume ejk 6= emand wm< Mk. Let ˜e1 be the (last) edge of ˜γ incident to m + 1. By definition of ˜γ, ˜e1 belongs to a path γ

′
l
vk v_{k}′
e_{jk}
em
˜
e1
e′
=m+1
γk
γ_{k}′
Ck
Figure 6: ˜e1 in γk′.
vl vk v_{k}′
e_{jk}
em
ˆ
e
e′
˜
e1
=m+1
γk
γ_{k}′
γ_{l}′
Ck
Cl
Figure 7: ˜e1 in γ
′
l with l < k.

in Cl with 1 ≤ l ≤ k as represented in Figure 6 for l = k and Figure 7 for

l < k. Note that if l < k then Cl, Cl+1, . . . , Ck−1, Ck are adjacent and by

Lemma 11 we have Ml= Mk. If w(˜e1) = Mk we take e = ˜e1. Otherwise we

have w(˜e1) < Mk. Let ˆe be the edge in E(Cl) \ {˜e1} incident to v

′

k. If l = k

then ˆe = emand we have w(ˆe) < Mk. Otherwise we apply Star condition to

{em, ˆe, ˜e1} and w(ˆe) < Mk is still satisfied. Then the Weak Cycle condition

applied to Cl implies that there is an edge e

′

in E(Cl) adjacent to ˜e1 with

w(e′) = Mk. If e

′

belongs to ˜γ we take e = e′. Otherwise, if |E(˜γ)| ≥ 2 there is an edge ˜e2 in ˜γ adjacent to e

′

and ˜e1 as represented in Figure 8.

Star condition applied to {e′, ˜e1, ˜e2} implies w(˜e2) = Mkand we take e = ˜e2.

Finally if |E(˜γ)| = 1 then vl = v1 = 1. Hence either e′ = e1 and the result

is satisfied or e′ is adjacent to e1 as represented in Figure 9. Then Star

condition applied to {e1, e

′

, ˜e1} implies w(e

′

can have e′ = ejlhence in the assumptions, Star condition has to be satisfied for G (and not only for T ).

vl v_{l}′
e_{jl}
ˆ
e
e′ ˜e2
˜
e1
=m+1
γl
γ′_{l}
Cl
Figure 8: |E(˜γ)| ≥ 2.
v1 v_{l}′
e1
e′
e_{jl}
ˆ
e
˜
e1
=m+1
γl
γ_{l}′
Cl

Figure 9: |E(˜γ)| = 1 and e′ 6= e1.

Case 2 Let us now consider Cl with l ≤ k and v

′

l 6= m + 1. Then there

exists an edge ˜e1∈ E(˜γ) \ E(Cl) incident to v

′

l. If l = k, then ˜e1 is the edge

of γk,m+1 incident to vk′ as represented in Figure 10.

vk v′_{k} m + 1
e_{jk}
˜
e1 em
γk
γ_{k}′
γk,m+1
Ck
Figure 10: ˜e1 in γk,m+1.

If l < k, we can assume that Cl and Cl+1 are not adjacent. Otherwise

Lemma 11 implies Ml= Ml+1 and we can replace Cl by Cl+1. If v

′

l 6= vl+1,

˜

e1 is the edge of γl,l+1 incident to v

′
l as represented in Figure 11.
vl v′_{l}
e_{jl}
˜
e1
γl
γ_{l}′
γl,l+1
Cl+1
Cl
Figure 11: v_{l}′ 6= vl+1, ˜e1 in γl,l+1.

If v_{l}′ = vl+1, ˜e1 is the first edge of γ

′

l+1 incident to v

′

l as represented in

Figure 12.

Let us denote by ˜γ1,l the part of ˜γ linking 1 to vl′. Let ˜e2 be the edge of ˜γ1,l

incident to v′_{l} as represented in Figure 13 with ˜e2 in γ

′

l. If ˜e2∈ γ/

′

vl v′_{l}
e_{jl} e_{jl+1}
˜
e1
γl
γ_{l}′
γl+1
γ_{l+1}′
Cl+1
Cl
Figure 12: v_{l}′ = vl+1, ˜e1 in γ
′
l+1.

exists a fundamental cycle Cr for some r, with 1 ≤ r ≤ l, such that ˜e2 ∈ γ

′

r

and Cr, Cr+1, . . . , Cl−1, Cl are adjacent. By Lemma 11 we have Mr = Ml.

Therefore we can suppose ˜e2∈ γl′ replacing Cl by Cr if necessary. Note that

in this last case Cl may be adjacent to Cl+1 but ˜e1 ∈ E(C/ l). Let ˆe be the

edge of γl incident to v
′
l as represented in Figure 13.
vl v_{l}′
e_{jl}
ˆ
e
˜
e2
˜
e1
γl
γ_{l}′
Cl
Figure 13: ˆe in γl, ˜e2 in γ
′
l.

If w(˜e2) = Ml, we take e = ˜e2. Otherwise we have w(˜e2) < Ml. If w(ˆe) = Ml,

Star condition applied to {ˆe, ˜e1, ˜e2} implies w(˜e1) = Ml and we can take

e = ˜e1. If now w(ˆe) and w(˜e2) < Ml, then the Weak Cycle condition

applied to Cl implies that there is an edge e

′

in E(Cl) adjacent to ˜e2 with

w(e′) = Ml. If e

′

belongs to ˜γ we take e = e′. Otherwise, if |E(˜γ1,l)| ≥ 2

there is an edge ˜e3 in ˜γ adjacent to e

′

and ˜e2 as represented in Figure 14.

Star condition applied to {e′

, ˜e2, ˜e3} implies w(˜e3) = Mland we take e = ˜e3.

Finally if |E(˜γ1,l)| = 1 then vl = v1 = 1. Then either e

′

= e1 and the

result is satisfied or e′ is adjacent to e1 as represented in Figure 15. Then

as w(˜e2) < w(e′), Star condition applied to {e′, e1, ˜e2} implies w(e′) = w1.

Note that the situation is similar to Case 1 with l < k replacing ˜e1 by ˜e2.

vl v′_{l}
e_{jl}
ˆ
e
e′ ˜e3
˜
e2
˜
e1
γl
γ_{l}′
Cl

Figure 14: |E(˜γ1,l)| ≥ 2, ˜e3 in ˜γ adjacent to e′ and ˜e2.

Using for instance Prim’s algorithm, we can find a spanning tree T of minimum weight in O(n2) time. The following proposition shows that the

v1 v_{l}′
e1
e′
e_{jl}
ˆ
e
˜
e2
˜
e1
γl
γ_{l}′
Cl

Figure 15: |E(˜γ1,l)| = 1, vl = v1= 1 and e

′ 6= e1.

Path condition is satisfied for G if it is satisfied for T and if all fundamental cycles associated with T satisfy the Weak Cycle condition, and therefore can be verified in polynomial time.

Proposition 13. *Let G = (N, E, w) be a weighted connected graph *
*satisfy-ing the Star condition. Let T be a minimum weight spannsatisfy-ing tree of G. If T*
*satisfies the Path condition and if each fundamental cycle associated with T*
*satisfies the Weak Cycle condition, then G also satisfies the Path condition.*

*Therefore the Path condition can be verified in O(n*3_{) time.}

*Proof. Let γ = {1, e*1, 2, e2, . . . , m, em, m + 1} be a path of G. We have to

prove that for every j, 2 ≤ j ≤ m − 1, we have wj ≤ max(w1, wm).

Let us consider e ∈ E(γ) \ E(T ) and C the fundamental cycle associated with T and e. Let M be the maximum edge weight in C. We necessarily have w(e) = M otherwise T is not a minimum weight spanning tree. We have to prove:

(6) M ≤ max(w1, wm).

Let ˜γ be the path in T related to γ linking 1 to m + 1 defined page 11. Lemma 12 implies that either M = w1 or wm and then (6) is obviously

satisfied or there exists ˜e ∈ E(˜γ) such that w(˜e) = M . Let e′_{1} (resp. e′_{m}) be
the edge of ˜γ incident to 1 (resp. m+1). Path condition applied to ˜γ implies
w(˜e) ≤ max(w_{1}′, w_{m}′ ). If e1 (resp. em) is not an edge of T , then it forms a

fundamental cycle containing e′_{1} (resp. e′_{m}) as represented in Figure 16 and
as T is a minimum weight spanning tree we have w_{1}′ ≤ w1 (resp. w

′

m ≤ wm).

If e1 (resp. em) is an edge of T , then Path condition applied to e1 ∪ ˜γ

(resp. em∪ ˜γ) implies w(˜e) ≤ max(w1, w

′

m) (resp. w(˜e) ≤ max(w

′

1, wm)). If

both e1 and em are edges of T , then Path condition applied to e1∪ ˜γ ∪ em

implies w(˜e) ≤ max(w1, wm). Hence in every case, w(˜e) ≤ max(w1, wm)

and therefore (6) is satisfied. As M is the maximum edge-weight in C, we get w(e) ≤ max(w1, wm) for all e ∈ E(C). Therefore we also have

w(e) ≤ max(w1, wm) for all e in a fundamental cycle associated with T .

Let e be a remaining edge in E(γ) ∩ E(T ). e is necessarily in ˜γ (more precisely in one of the subpaths γ0,1, γ1,2, . . . , γk,m+1) and therefore satisfies

w(e) ≤ max(w1, wm).

Let us now analyze the complexity. Prim’s algorithm provides a minimum
weight spanning tree in time O(n2_{).}

1 m
e1 em
e′_{m}
e′1
γ
˜
γ

Figure 16: γ and ˜γ, e1 ∈ E(T ), e/ m∈ E(T ).

Then we verify Path condition for T . For each vertex v ∈ V , we build
the arborescence in T rooted in v. It can be done using a breadth first search
in O(n2). During the search we can verify the Path condition. Indeed we
only have to compare the weight of an edge added to the arborescence to
the weight of its predecessor in the arborescence. If there is an increase of
edge-weights followed by a decrease then it contradicts the Path condition
(We can assign a label {+, −} to each vertex entering the arborescence, and
therefore associated with the entering edge, corresponding to an increase or
decrease, and we compare it to the label of the preceding node). Hence we
can verify the Path condition on T in O(n3_{) time.}

For a given fundamental cycle associated with T , the Cycle condition can be verified in O(n) time. Indeed, as T contains at most n − 1 edges, a fundamental cycle contains at most n edges. Then an edge of minimum weight can be found after at most n − 1 comparisons. An edge of minimum weight among the n − 1 remaining edges can be found using at most n − 2 comparisons. Actually we only have to consider the two edges adjacent to the minimum weight edge previously found. Finally we only have to verify the equality of edge-weights for the remaining edges. It requires at most n − 3 comparisons. There exists m − (n − 1) edges in E \ E(T ) and therefore m − (n − 1) fundamental cycles associated with T . Hence we can verify the Cycle condition for all fundamental cycles associated with T in O(nm).

Finally Path condition for G can be verified in O(n2 _{+ n}3_{+ nm) time}

and therefore in O(n3) time (as m ≤ n(n−1)_{2} ).

We now prove that the Adjacent cycles condition can be verified in O(n6) time. We first establish that we only need to verify the condition on specific cycles built with shortest paths.

Lemma 14. *Let G = (N, E, w) be a weighted graph satisfying the Star and*

*Path (and therefore Weak and Intermediary Cycle) conditions. Let e*1 =

*{1, 2} and e*2 *= {2, 3} be two adjacent edges of G. Let us define ˜*E := {e ∈

E; w(e) > max(w1, w2*)} and ˜*G := (N \ {2}, ˜E, w_{| ˜}E*). Let γ be a shortest*

*path in ˜G, if it exists, linking 1 and 3. Let us consider the following claims:*

*1) There exist two simple cycles C and C*′ *in G satisfying conditions a, b,*

*(e) e*1 *and e*2 *are two common non-maximum weight edges of C and C*

′

*.*

*2) There exist two simple cycles ˜C and ˜*C′ *satisfying conditions a, b, c, d of*

*the Adjacent Cycles condition, and such that:*

*(f ) ˜C and ˜*C′ *have two common non-maximum weight edges ˜*e1*= {˜1, 2},*

˜

e2= {2, ˜*3}.*

*(g) ˜C \ {2} corresponds to a shortest path ˜γ in ˜G linking ˜1 and ˜3. ˜γ is*

*a subpath of γ and for some i ∈ V (˜γ) \ {˜1, ˜3}, ˜*C′*\ {2} corresponds*

*to a shortest path γ*′

*in ˜*G_{N\{2,i}} *linking ˜1 and ˜3.*

*Then Claim 1 implies Claim 2. Moreover for given adjacent edges e*1 *and*

e2 *of G, Claim 2 can be verified in O(n*3*) time.*

Remark 2. It can happen that ˜γ = γ and then 1 = ˜1 and 3 = ˜3.

Remark 3. Let us observe that |V (C)| ≥ 4 and |V (C′)| ≥ 4, otherwise by the Weak Cycle condition C or C′ would have a maximum weight chord. Let us also observe that by the Weak Cycle condition and Lemma 7 page 8, all edges of γ and γ′ have the same weight M .

˜ 3 2 i ˜1 M M M ˜ w1 ˜ w2 M M M M M ˜ C′ ˜ C Figure 17: i ∈ ˜γ = ˜C \ {2} and i 6∈ γ′ = ˜C′\ {2}.

*Proof of Lemma 14. We assume w.l.o.g. w*1 ≤ w2. Let us assume that

Claim 1 is satisfied. Lemma 7 implies max_{e∈ ˆ}_{E(C)}w(e) = max_{e∈ ˆ}_{E(C}′_{)}w(e) =
M . Let γ be a shortest path in ˜G linking 1 to 3. Then ˆC = {2, e1, 1} ∪ γ ∪

{3, e2, 2} is a simple cycle in G adjacent to C or ˆC = C. The Weak Cycle

condition and Lemma 7 imply that e1 and e2 are edges of non-maximum

weight in ˆC and all edges of γ have weight M = max_{e∈ ˆ}_{E(C)}w(e). As M >
w2 ≥ w1, the Intermediary Cycle condition implies w(e) = M for all e ∈

ˆ

E( ˆC) non incident to 2 and w(e) = w2 or w(e) < w1 = w2 for any chord

incident to 2. Hence ˆC has no chord of maximum weight M otherwise γ is not a shortest path.

Case 1 Let us assume that ˆC has no chord. Then we take ˜C = ˆC. If V ( ˜C) \ V (C′) 6= ∅ we select a node i in V ( ˜C) \ V (C′). Otherwise we have V ( ˜C) ⊆ V (C′). V ( ˜C) ⊂ V (C′) would imply that C′ has a chord of maximum weight M , a contradiction. Hence V ( ˜C) = V (C′) and in this case we have V ( ˜C) \ V (C) = V (C′) \ V (C) 6= ∅. Then we select a node i in

V ( ˜C) \ V (C) (we can interchange C and C′). By construction of i there
exists a path in ˜GN\{2,i} *linking 1 to 3 (i.e., a path in C*

′

or C). Therefore there exists a shortest path γ′ in ˜GN\{2,i} linking 1 to 3. Then the cycle

˜

C′ = {2, e1, 1} ∪ γ′∪ {3, e2, 2} is convenient. We have V ( ˜C) \ V ( ˜C′) 6= ∅ as

i ∈ V ( ˜C) \ V ( ˜C′). If V ( ˜C′) \ V ( ˜C) = ∅ then V ( ˜C′) ⊆ V ( ˜C). V ( ˜C′) = V ( ˜C)
is not possible as i ∈ V ( ˜C) \ V ( ˜C′). Then V ( ˜C′) ⊂ V ( ˜C) and γ′ is a path
strictly shorter than γ linking 1 to 3 in ˜G, a contradiction. ˜C and ˜C′ have no
maximum weight chord because γ and γ′ are shortest path linking 1 and 3
in respectively ˜G and ˜G_{N\{2,i}}. ˜C has no chord. ˜C and ˜C′ satisfy conditions
a to d, 2f and 2g.

If e1 and e2 are given, we can obtain a shortest path γ, if it exists, using

a Breadth First Search algorithm in O(n2_{) time. Then to decide if ˆ}_{C has}

a chord or not, take O(n) time. Indeed ˆC cannot have a maximum weight chord otherwise γ is not a shortest path, and any non maximum weight chord is incident to 2. Therefore we only have to check in G if there exists a neighbor of 2 in γ \ {1, 3}. In the worst case, we have to consider n − 3 vertices. For a given node i ∈ γ \ {1, 3} checking the existence of {2, i} in the adjacency matrix is in O(1) time. Therefore we can check the existence of a chord in O(n) time. If ˆC is chordless then for a given i ∈ V (γ) we look for a shortest path γ′ in ˜GN\{2,i}linking 1 and 3 using the BFS algorithm in

O(n2) time. In the worst case we repeat this for all i ∈ V (γ). Hence in this
case ( ˆC without chord) we need O(n3_{) time to decide if Claim 2 is satisfied.}

Case 2 Let us now assume that ˆC has a chord ˆe. Then it is neces-sarily a non-maximum weight chord incident to 2 with w(ˆe) ≤ w2 (by

the Intermediary Cycle condition). We choose arbitrarily the end-vertex i of any chord ˆe0 = {2, i} of ˆC. We select two other edges in ˆE( ˆC),

ˆ

e1 = {2, k} and ˆe2 = {2, j}, with 3 ≤ j < i < k, which are as close

as possible from ˆe0*, i.e., j is the maximum possible index for such an*

edge ˆe2 and k is the minimum possible index for such an edge ˆe1. Hence

if j = 3 (resp. k = 1) then ˆe2 = e2 (resp. eˆ1 = e1). We consider

the cycle ˜C = {k, ˆe1, 2, ˆe2, j, ej, j + 1, . . . , ei−1, i, ei, . . . , ek−1, k} as

repre-sented in Figure 18. By renumbering as follows, ˜1 = k, ˜e1 = ˆe1, ˜2 = 2,

3 j 2 i 1 k e2 e1 ˆ e0 ˆ e2 ˆ e1 ˜ C ˆ C Figure 18: ˆC and ˜C. ˜

e2 = ˆe2, ˜3 = j, ˜e_{˜i−1} = ei−1, ˜i = i, ˜e˜i = ei, . . ., ˜em = ek−1, k = ˜1

maximum weight chord ˜e0 = {2,˜i} corresponding to ˆe0 = {2, i} and ˜e1,

˜

e2 are the two non-maximum weight edges of ˜C. The restricted path ˜γ =

{˜3, ˜e3, . . . , ˜e_{˜i−1},˜i, ˜e˜i, . . . , ˜em, ˜1} is a shortest path linking ˜3 to ˜1 in ˜G as γ is

a shortest path linking 1 to 3 in ˜G. As C and C′ have no common chord, ˜

e0 = {2,˜i} cannot be a common chord of C and C

′

. We can assume w.l.o.g. that ˜e0 = {2,˜i} is not a chord of C′. Let γ1 (resp. γ2) be the part of

γ linking 1 and k (resp. 3 and j), γ1 = {k, ek, k + 1, . . . , p, ep, 1} (resp.

γ2 = {3, e3, 4, . . . , j − 1, ej−1, j}) then γ1 ∪ (C

′

\ {2}) ∪ γ2 corresponds to

a path linking k to j and therefore ˜1 to ˜3 in ˜G \ {2,˜i} as represented in Figure 19. Therefore there exists a shortest path γ′ in ˜G \ {2,˜i} linking ˜1 to

3 ˜3
2 ˜_{i}
1 ˜1
˜
e2
˜
e0
˜
e1
e1
e2
C′
Figure 19: Path {˜1, 1} ∪ (C′ \ {2}) ∪ {3, ˜3}.
˜

3. Then we can consider the cycles ˜C and ˜C′ = {2, ˜e1, ˜1} ∪ γ

′

∪ {˜3, ˜e2, 2} as

represented in Figure 20. We can achieve the proof as in the first case. By

˜
3
2 ˜_{i}
˜
1
˜
e2
˜
e0
˜
e1
˜
C
˜
C′

Figure 20: Adjacent cycles ˜C and ˜C′ .

construction ˜C has only one non-maximum weight chord ˜e0 = {2,˜i} which

is not a chord of ˜C′

(˜i /∈ ˜C′

) and ˜C and ˜C′

have no maximum weight chords. ˜

C and ˜C′ satisfy conditions a to d, 2f and 2g (replacing e1 (resp. e2) by ˜e1

(resp. ˜e2)).

As in Case 1, we can obtain ˆC, if it exists, using a Breadth First Search algorithm in O(n2) time. Then to find a chord ˜e0 = {2,˜i} of ˆC, we need at

most O(n) time. To select the edges ˜e1 = {2, k} and ˜e2 = {2, j}, we also

need at most O(n) comparisons. To build the shortest path γ′ in ˜G \ {2,˜i} using a Breadth First Search algorithm, we need O(n2) time. Hence to verify Claim 2 when ˆC has a chord, we need O(n2) time. Note that if there exists a chord in ˜C we do not need to check Claim 2g for all i ∈ V (˜γ) \ {˜1, ˜3}. We check the condition only for the node i corresponding to the chord we have found.

We consider that the Adjacent cycles condition can be divided in two parts. The first part is the condition of non-existence of cycles with two

common non-maximum weight edges. Then the second part is the condition on edge-weights for cycles with a unique common non-maximum weight edge.

Proposition 15. *The first part of the Adjacent cycles condition can be*

*verified in O(n*6*) time.*

*Proof. We have to verify that for any pair of adjacent edges e*1 = {1, 2} and

e2 = {2, 3}, e1 and e2 are not common non-maximum weight edges of two

*adjacent cycles (i.e., Claim 1 of Lemma 14 is not satisfied). By Lemma 14*
it is sufficient to prove that Claim 2 of Lemma 14 is not satisfied and it
can be done in O(n3_{) time. There are at most} P

i∈V Cn2 = n2 n−12 pairs

of adjacent edges. Hence we need at most n3 _{iterations of the procedure}

considered in Lemma 14 and therefore the first part of the Adjacent cycles
condition can be verified in O(n6_{) time.}

Lemma 16. *Let G = (N, E, w) be a weighted graph satisfying the Star and*
*Path (and therefore Intermediary Cycle) conditions. Let us assume that the*
*first part of the Adjacent cycles condition is satisfied. Then the following*
*Claims are equivalent:*

*1) There exist two simple cycles C and C*′ *in G satisfying conditions a, b,*

*c, d of the Adjacent Cycles condition, and such that:*

*(e) C and C*′ *have a unique common non-maximum weight edge e*1 =

*{1, 2} and w(e) > w*1 *for all e ∈ (E(C) ∪ E(C*

′

)) \ {e1*}.*

*2) There exist two simple cycles ˜C and ˜*C′ *in G satisfying conditions a, b,*

*c, d of the Adjacent Cycles condition, and such that:*

*(f ) ˜C and ˜*C′ *have a unique common non-maximum weight edge ˜*e1 =

*{1, 2} and w(e) > w*1 *for all e ∈ (E( ˜*C) ∪ E( ˜C

′

)) \ {e1*}.*

*(g) ˜*C\{˜e1*} corresponds to a shortest path ˜γ in ˜*G = (V, E\{˜e1}, w|E\{˜e1})

*linking 1 and 2 and for some i ∈ V (˜γ) \ {1, 2}, ˜*C′\ {˜e1*} corresponds*

*to a shortest path ˜*γ′ *in ˜*GN\{i} *linking 1 and 2.*

p _{p}′
1
i 2
3 3′
˜
e_{p−1} ˜ep
˜
e2
˜
e1
˜
e′_{p}
˜
e′2
˜
C C˜′
Figure 21: ˜C = {1, ˜e1, 2, ˜e2, . . . , p, ˜ep, 1} and ˜C
′
= {1, ˜e1, 2, ˜e
′
2, . . . , p
′
, ˜e′
p′, 1}.

Let us observe that using the Intermediary Cycle condition we only need to verify that the edges in E(C) adjacent to e1 have a weight strictly greater

than w1. Indeed if it is satisfied then w(e) > w1 for all e ∈ E(C) \ {e1}.

Remark 4. Let us note that by the Intermediary Cycle condition C, C′, ˜C, ˜C′ can have at most two maximum weight edges. If they all have two non-maximum weight edges then e1 and all these edges are adjacent (incident to

1 or 2).

*Proof. Let us consider C = {1, e*1, 2, e2, . . . , m, em, 1}. Then γ = C \ {e1} =

{2, e2, . . . , m, em, 1} is a path linking 1 and 2 in ˜G. Hence there is a shortest

path ˜γ = {2, ˜e2, ˜3, ˜e3, . . . , p, ˜ep, 1} linking 1 and 2 in ˜G. Then we can build

the cycle ˜C := {1, e1, 2, ˜e2, ˜3, ˜e3, . . . , p, ˜ep, 1} adding the edge e1 to the path

˜

γ. Either ˜C = C, or ˜C and C are adjacent. In this last case Lemma 7 implies
max_{e∈E(C)}w(e) = max_{e∈E( ˜}_{C)}w(e). Hence e1 is a non-maximum weight edge

in ˜C. Claim 1e implies w2 > w1 (resp. wm > w1). If ˜e2 = e2 (resp ˜ep= em)

then w(˜e2) = w2 (resp. w(˜ep) = wm). Otherwise Star condition applied

to {e1, e2, ˜e2} (resp. e1, em, ˜ep) implies w(˜e2) = w2 (resp. w(˜ep) = wm).

Then the Intermediary Cycle condition implies w1< w(˜e2) ≤ w(˜e3) = . . . =

w(˜ep) = M with M = max_{e∈E( ˜}_{C)}w(e), exchanging ˜e2 and ˜ep if necessary.

The Intermediary Cycle condition also implies that for any chord e in ˜C, we have w(e) ≤ w2 if e is incident to 2 (in fact, as w1 < w2, the Star condition

applied to {e1, ˜e2, e} implies w(e) = w2 ) and w(e) = M otherwise. In any

case e would contradict the optimality of ˜γ. Hence ˜C has no chord.

Let us assume V ( ˜C) ⊆ V (C) ∩ V (C′). As V (C) \ V (C′) 6= ∅ (resp. V (C′) \ V (C) 6= ∅) we have V ( ˜C) 6= V (C) (resp. V ( ˜C) 6= V (C′)). Then at least one edge ˜e of ˜C is a chord of C. As C has no maximum weight chord, ˜

e is a chord of non-maximum weight in C and then ˜e is also an edge in E( ˜C) \ {e1} of non-maximum weight (by Lemma 7, as C and ˜C are adjacent

the maximum weight in C and ˜C is M ). Following the Intermediary Cycle condition ˜e is the unique edge of ˜C of non-maximum weight adjacent to e1

as represented in Figure 22. By the same reasoning (interchanging C and

e1 w1 e2 w2 M M M M ˜ e C ˜ C

Figure 22: ˜e non-maximum weight chord (resp. edge) in C (resp. ˜C). C′), there exists an edge ˜e′ of ˜C which is a chord of C′ and ˜e′ is an edge of

˜

C of non-maximum weight. ˜e′ necessarily corresponds to the unique edge of ˜

C of non-maximum weight adjacent to e1. Therefore ˜e = ˜e

′

. Hence C and C′ have a common chord, a contradiction.

Therefore we have either V ( ˜C) \ V (C) 6= ∅ or V ( ˜C) \ V (C′) 6= ∅.
In every case we can choose i ∈ V ( ˜C) \ {1, 2} such that there exists a
path ˆγ in ˜G_{N\{i}} linking 1 and 2. γ = γ if i /ˆ ∈ V (C) or ˆγ = C′ \
{e1} = {2, e
′
2, 3
′
, e′_{3}, . . . , m′, e′_{m}′, 1} if i /∈ V (C
′
) as represented in
Fig-ure 23. Therefore we can find a shortest path ˜γ′ in ˜G_{N\{i}} linking 1 and
2 as represented in Figure 21. Adding to ˜γ′ the edge e1 we get the

cy-cle ˜C′ = {1, e1, 2, ˜e′2, ˜3
′
, ˜e′_{˜}_{3}′, . . . , ˜p
′
, ˜e′_{p}_{˜}′, 1}. C˜
′

has no chord otherwise ˜γ′
is not a shortest path in ˜G_{N\{i}} (by the same reasoning as for ˜C). As
i ∈ V ( ˜C) \ V ( ˜C′) we have V ( ˜C) \ V ( ˜C′) 6= ∅. We have already seen that
max_{e∈E(C)}w(e) = max_{e∈E( ˜}_{C)}w(e) = max_{e∈E( ˜}_{C}′_{)}w(e), and that if C
con-tains a second non-maximum weight edge then ˜C and ˜C′

also contain a sec-ond non-maximum weight edge of same weight. Therefore if V ( ˜C′) ⊂ V ( ˜C) then ˜γ′ is a path linking 1 and 2 in ˜G shorter that ˜γ, a contradiction. There-fore we have V ( ˜C′) \ V ( ˜C) 6= ∅. 3′ 2 3 1 ˜3 m′ i ˜ e2 ˜ e0 e1 e′2 C′ C˜ Figure 23: C′ and ˜C, with i /∈ C′ .

We first verify a weaker condition than the second part of the Adjacent Cycles condition.

Weak Second Part of the Adjacent Cycles Condition. If e1

is a unique non-maximum weight edge common to two simple cycles (C, C′) in G satisfying the conditions a, b, c, d of the Adjacent Cycles condition, then there are non-maximum weight edges e2 ∈ E(C) \ {e1}

and e′

2∈ E(C

′

) \{e1} such that e1, e2, e′2 are adjacent and w1 = w2≥ w2′

or w1 = w

′

2≥ w2.

Note that we cannot have e2 = e

′

2 otherwise e1 and e2 are two

non-maximum weight edges common to C and C′ as w1 = w2 = w

′

2. Therefore

we have e2 ∈ E(C) \ E(C′) and e′2∈ E(C

′

) \ E(C).

Proposition 17. *Let G = (N, E, w) be a weighted graph satisfying the Star*
*and Path (and therefore Weak and Intermediary Cycle) conditions, and the*
*first part of the Adjacent Cycles condition. Then the Weak Second Part of*

*Proof. Let e*1be a common non-maximum weight edge of two adjacent cycles

C and C′ satisfying the conditions a, b, c, d of the Adjacent Cycles condition. Let e2 ∈ E(C) \ e1 and e

′

2 ∈ E(C

′

) \ e1 be edges incident to the same

end-vertex of e1 (at this step of the proof e2 may be equal to e

′

2 and e2 or e

′

2

may have maximum weight in C or C′). Let us assume that we have neither w1 = w2 ≥ w ′ 2 nor w1 = w ′ 2 ≥ w2. If e2 = e ′ 2 then w2 = w ′ 2 and therefore we have w1 < w2 = w ′ 2 or w1 > w2 = w ′

2. This last case is not possible as

e1, e2 would be two non-maximum weight edges common to C and C′. If

e2 6= e

′

2 then Star condition applied to e1, e2, e

′

2 also implies w1 < w2 = w

′

2.

(By the same reasoning we get w1 < wm = w

′

m′.) Then the Intermediary

Cycle condition implies w(e) > w1 for all e ∈ (E(C) ∪ E(C′)) \ {e1}. Then

by Lemma 16, e1 is also a non-maximum weight edge common to two cycles

˜

C and ˜C′ such that ˜C \ {e1} corresponds to a shortest path ˜γ linking 1 and

2 in ˜G = (N, E \ {e1}, w|E\{e1}) and for some i ∈ V (˜γ) \ {1, 2}, ˜C ′

\ {e1}

corresponds to a shortest path in ˜G_{N\{i}} linking 1 and 2. Moreover we still
have w(e) > w1 for all e ∈ (E( ˜C) ∪ E( ˜C

′

)) \ {e1}. Therefore for every

edge e1 ∈ E, we only have to verify that such a couple of adjacent cycles

( ˜C, ˜C′) cannot exist. Therefore we look for a shortest path ˜γ linking 1 and
2 in ˜G. If it exists then we look for a shortest path ˜γ′ in ˜G_{N\{i}} with
i ∈ V (˜γ) \ {1, 2}, linking 1 and 2. If such a path ˜γ′ exists, we only have to
verify that w(e) = w1 for some edge e of ˜γ or ˜γ′ (in fact the Weak Cycle

condition implies that such an edge is necessarily adjacent to e1).

Note that by the Star and Weak Cycle conditions and Lemma 7, all elementary paths linking 1 and 2 in ˜G have either all their edges of same weight M > w1 or one edge incident to 1 or 2 with weight w2 > w1 and

their remaining edges with weight M > w2. Therefore we can find a shortest

path ˜γ linking 1 and 2 in ˜G using a BFS algorithm in O(n2). Then we can also use a BFS algorithm to find a shortest path in ˜GN\{i} with i ∈ ˜γ. In

the worst case we have to consider all nodes i ∈ ˜γ. Therefore we can verify
in O(n3_{) time if e}

1 satisfies the Weak Second Part of the Adjacent Cycles

condition. Therefore the Weak Second Part of the Adjacent cycles condition
can be verified in O(n3_{m) time.}

Proposition 18. *Let G = (N, E, w) be a weighted graph satisfying the*
*Star and Path (and therefore Intermediary Cycle) conditions, and the Weak*
*Second Part of the Adjacent Cycles condition. Then the Cycle condition is*
*satisfied.*

*Proof. Let us consider a cycle C = {1, e*1, 2, e2, . . . , m, em, 1}. The

Interme-diary Cycle condition implies that w1 ≤ w2 ≤ w3 = · · · = wm = M after

renumbering if necessary and that w(e) ≤ w2 for all chord incident to 2 and

w(e) ≤ M for all other chords. Moreover:

to 2. If e is a chord incident to 2 then w1 ≤ w2 = w(e) < M or

w(e) < w1= w2 < M .

• If w1< w2 = M , then w(e) = M for all e ∈ E(C) \ {e1}.

Therefore if w1 < w2 = M the Cycle condition is satisfied. If w1 ≤ w2 <

M let us assume by contradiction that there exists a chord incident to 2 with w(e) < w2. Star condition applied to {e, e1, e2} implies w1 = w2

and that for any other chord e′ incident to 2 we have w(e′) = w1 = w2.

Hence we can assume that C has only one chord e incident to 2 replacing if necessary C by a smaller cycle using the chords of C incident to 2 which are as close as possible to e. We have e = {2, j} with 4 ≤ j ≤ m. We can consider two adjacent cycles ˜C := {2, e, j, ej, . . . , m, em, 1, e1, 2} and

˜ C′ := {2, e2, 3, e3, . . . , ej−1, j, e, 2} as represented in Figure 24. j m 1 3 2 M M M e1 w1 e2 w2 M e ˜ C C˜′ Figure 24: w(e) < w1= w2.

e is a unique common edge to ˜C and ˜C′ such that w(e) < w1 = w2 ≤ M .

If ˜C (resp. ˜C′) has a chord then we can replace ˜C (resp. ˜C′) by a smaller cycle. Therefore we can suppose that ˜C and ˜C′ have no chords. Then w(e) < w1 = w2 contradicts the Weak Second Part of the Adjacent Cycles

condition. Therefore any edge incident to 2 has weight w2.

Finally if w1 = w2 = M , let us also assume by contradiction that there

exists a chord e in C with w(e) < w2 = M . As w1 = w2 = M we can

always assume e incident to 2 after renumbering if necessary and then we can establish the same contradiction as before.

Assuming now that the Star, Path, and Cycle conditions are satisfied we give a new formulation of the Pan condition. Then we will analyze its complexity.

Lemma 19. *Let G = (N, E, w) be a connected weighted graph satisfying the*
*Star, Path, and Cycle conditions. The Pan condition is satisfied if and only*

*if for all simple cycle C = {1, e*1, 2, e2, . . . , m, em*, 1}, with w*1 = w2 ≤ w3 =

· · · = wm = ˆM = max_{e∈ ˆ}_{E(C)}*w(e), one of the following claim is satisfied:*

*1) w*1= w2 *= M .*

*3) σ(E) = w*1 = w2*< M , where σ(E) = min*e∈E*w(e).*

*Proof. Let us first assume that the Pan condition is satisfied. Let us consider*

a simple cycle C = {1, e1, 2, e2, . . . , m, em, 1}, with w1 = w2 ≤ w3 = · · · =

wm = ˆM . If w1 = w2 = ˆM then Claim 1 is satisfied. Otherwise we have

w1 = w2 < ˆM . If {1, 3} is a chord of C then Claim 2 is satisfied. Otherwise

{1, 3} is not a chord of C and let us assume σ(E) < w1 = w2. Then there

exists e ∈ E \ E(C) such that w(e) < w1 = w2. Note that the Cycle

condition implies that a chord of C incident (resp. non incident) to 2 has weight w2 (resp. ˆM ). Therefore we actually have e ∈ E \ ˆE(C). As G is

connected there is a path P containing e (P can be restricted to e) such that |V (C) ∩ V (P )| = 1. Then as w(e) < w1 = w2 < M , Pan condition implies

that V (C) ∩ V (P ) = {2} and that {1, 3} is a maximum weight chord of C, a contradiction. Therefore Claim 3 is satisfied.

Let us now consider a simple cycle C = {1, e1, 2, e2, . . . , m, em, 1}, and

an elementary path P such that there is an edge e ∈ P with w(e) ≤ min1≤k≤mwk and |V (C) ∩ V (P )| = 1. The Cycle condition implies w1 ≤

w2 ≤ w3 = · · · = wm = ˆM after renumbering if necessary. Moreover the

Path and Star conditions imply w1 = w2 ≤ ˆM and if w1 = w2 < ˆM then

V (C) ∩ V (P ) = {2} (as the Weak Pan condition is satisfied by Proposi-tion 6). If C satisfies Claim 1 then the Pan condiProposi-tion is trivially satisfied. If C satisfies Claim 2 then {1, 3} is a chord of C and as w1 = w2< ˆM we have

V (C) ∩ V (P ) = {2}. Moreover the (Intermediary) Cycle condition implies that {1, 3} is a maximum weight chord. Therefore the Pan condition is sat-isfied. Finally if C satisfies Claim 3, we also have V (C) ∩ V (P ) = {2} and as σ(E) = w1, we necessarily have w(e) = w1. Therefore the Pan condition

is still satisfied.

Proposition 20. *Let G = (N, E, w) be a connected weighted graph *
*satisfy-ing the Star, Path conditions and the Cycle condition (or the Weak Second*
*Part of the Adjacent Cycles condition). Then Pan condition can be verified*

*in O(n*5*) time.*

*Proof. Let C = {1, e*1, 2, e2, . . . , m, em, 1} be a simple cycle of G. The Cycle

condition implies w1 ≤ w2 ≤ w3 = · · · = wm = M = maxe∈E(C)w(e),

after renumbering if necessary. Let ˜γ be a shortest path linking 1 and
3 in G_{N\{2}} and let ˜C be the simple cycle formed by e1, e2 and ˜γ. If

˜

C 6= C then Lemma 7 implies that max_{e∈E( ˜}_{C)} = M . Lemma 19 implies
that Pan condition for C is equivalent to Pan condition for ˜C. Then if
w1 = w2 *< M either the chord {1, 3} exists (i.e. ˜*γ = {1, {1, 3}, 3} and

˜

C = {1, e1, 2, e2, 3, {1, 3}, 1}) otherwise w1= w2 = σ(E). Hence to verify the

Pan condition we just need to consider any pair of adjacent edges e1= {1, 2}

and e2 = {2, 3} with w1 = w2. Let us consider the graph ˜G = (N, ˜E, w_{| ˜}_{E})

linking 1 to 3 in the graph ˜G_{N\{2}} using a BFS algorithm in O(n2_{) time.}

If γ exists and if γ has at least two edges, we consider the simple cycle C = {1, e1, 2, e2, 3} ∪ γ. Note that as γ is a shortest path linking 1 and

3, {1, 3} cannot be a chord of ˜C. Then applying Lemma 19 to ˜C we only need to verify that either w1 = w2 = w3 or w1 = w2 = σ(E). σ(E) can

be computed in O(m) time. We have at most P

i∈V Cn2 = n2 n−12 pairs

e1, e2 to take into account. Therefore the Pan condition can be verified in

O(n5).

Proposition 21. *Let G = (N, E, w) be a weighted graph satisfying the Weak*
*Second Part of the Adjacent Cycles and the Pan conditions. Then the Second*
*part of the Adjacent Cycles condition is satisfied.*

*Proof. Let e*1be a unique non-maximum weight edge common to two simple

cycles (C, C′) in G satisfying the conditions a, b, c, d of the Adjacent Cycles condition. The Weak Second Part of the Adjacent Cycles condition implies that there are non-maximum weight edges e2 ∈ E(C)\{e1} and e

′ 2 ∈ E(C ′ )\ {e1} such that e1, e2, e ′

2 are adjacent and w1 = w2 ≥ w

′

2 or w1 = w

′

2 ≥ w2.

If |E(C)| = 3 or |E(C′

)| = 3 then the second part of the Adjacent Cycles condition is satisfied. If |E(C)| ≥ 4 and |E(C′)| ≥ 4, let us assume w1 =

w2 > w ′ 2 (resp. w1 = w ′ 2 > w2). By assumption C (resp. C ′ ) have no maximum weight chord therefore the Pan condition applied to C and e′

2

(resp. C′ and e2) implies w1 = w2 = maxe∈E(C)w(e) (resp. w1 = w′2 =

max_{e∈E(C}′_{)}w(e)), a contradiction. Therefore w_{1}= w_{2} = w
′

2 and the second

part of the Adjacent Cycles condition is satisfied.

Theorem 22. *Let G = (N, E, w) be an arbitrary weighted graph. Let us*

*consider the correspondance P*min*. We can verify inheritance of F-convexity*

*from (N, v) to (N, v) for all F-convex and superadditive game (N, v) in*

O(n6*) time.*

### 5

### Conclusion

We think that the method we have presented to prove that inheritance of F-convexity is a polynomial problem, especially, the use of minimum spanning trees and shortest paths to select specific paths and cycles, can be useful for further developments. In particular it could be used to obtain polynomial algorithms to check inheritance of convexity for Pmin or of convexity or

F-convexity for other correspondences as for example the correspondence PG

presented in (Grabisch and Skoda, 2012) which associates to a subset its partition into relatively strongest components. Of course it is not certain that such algorithms do exist.

### Acknowledgments

The support of the French National Research Agency, through DynaMITE project (Reference: ANR-13-BSH1-0010-01), is gratefully acknowledged.

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